Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where c i is any point in the ith subinterval. lim ‖ Δ ‖ → 0 ∑ i = 1 n ( 1 + 3 c i ) Δ x i , [ 1 , 5 ]
Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where c i is any point in the ith subinterval. lim ‖ Δ ‖ → 0 ∑ i = 1 n ( 1 + 3 c i ) Δ x i , [ 1 , 5 ]
Solution Summary: The author explains the formula used to calculate the definite integral of f(x).
Writing a Limit as a Definite Integral In Exercises 11-14, write the limit as a definite integral on the given interval, where
c
i
is any point in the ith subinterval.
lim
‖
Δ
‖
→
0
∑
i
=
1
n
(
1
+
3
c
i
)
Δ
x
i
,
[
1
,
5
]
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Prove the Cauchy Mean Value Theorem. Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not equal to h(b). There exists c in (a, b) such that (f(b) - f(a))h'(c) = f'(c)(h(b)-h(a)).
math
Prove that f(x) = x ⋅ |x| is continuous at all points c in ℝ.
Fermat’s Theorem:
Let f be a continuous function on an open interval I where c is in I .
If f has an extremum at c, then c must be a critical number of f.
Expound this theorem.
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